Cylindrical pump-loss balance (${\int }_S{n}^{2}dS=(\gamma /\Gamma ){\int }_{S}ndS$)

For a stationary pumped condensate in a closed cross-section, integrate the gain-loss Madelung equations. Vanishing lateral condensate number current and constant integrated axial current imply the displayed balance. It fixes a density-weighted mean, not the pointwise density. If the integrated axial current $J(z)$ varies, the exact relation instead is ${\int }_S{n}^{2}dS=(\gamma /\Gamma )N(z)-[\hslash /(2\Gamma )]J^{\prime }(z)$. Thus lateral no-flux conditions alone are insufficient for a slice-by-slice balance in an axially inhomogeneous state.